First picture, move on to HARD you mean ?
Usually dungeons in closers have to complete all easy & normal ones first then you can move on to hard ones. Or maybe your equipment need to be fix.
Second picture, i have the same problem with you man, i dont know why icant equip it, i think thats a quest item or something we must have so we can travel between town.
Im a leve 27 newbie so... this is my opinion
Eh, maybe someone else can help you out more, since I haven't played Closers in awhile..
The first one means you don't have enough stamina (That's what the 10 means, you need 10 stamina per dungeon)
The second one is a quest item
2nd item is a quest costume that should either be tunned,enhanced or dismantled,if you got additional item with it then its probably tunning,if not then just dismantle it.
Check costume/tunning guide for more details.